Theorem f1omo 48751

Description: There is at most one element in the function value of a constant function whose output is 1_o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 48750 assuming ax-un 7737 (see f1omoALT 48752). (Contributed by Zhi Wang, 19-Sep-2024.)

Hypothesis

Ref	Expression
f1omo.1	⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))

Assertion

Ref	Expression
f1omo	⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑋   𝜑,𝑦

Proof of Theorem f1omo

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1oex 8498	. . . 4 ⊢ 1o ∈ V
2		eqid 2734	. . . 4 ⊢ ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋)
3	1, 2	fvconst0ci 48749	. . 3 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o)
4		mo0 48691	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
5		el1o 8515	. . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅)
6		el1o 8515	. . . . . . . 8 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅)
7		eqtr3 2756	. . . . . . . 8 ⊢ ((𝑦 = ∅ ∧ 𝑥 = ∅) → 𝑦 = 𝑥)
8	5, 6, 7	syl2anb 598	. . . . . . 7 ⊢ ((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥)
9	8	gen2 1795	. . . . . 6 ⊢ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥)
10		eleq1w 2816	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 1o ↔ 𝑥 ∈ 1o))
11	10	mo4 2564	. . . . . 6 ⊢ (∃*𝑦 𝑦 ∈ 1o ↔ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥))
12	9, 11	mpbir 231	. . . . 5 ⊢ ∃*𝑦 𝑦 ∈ 1o
13		eleq2w2 2730	. . . . . 6 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ 𝑦 ∈ 1o))
14	13	mobidv 2547	. . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ ∃*𝑦 𝑦 ∈ 1o))
15	12, 14	mpbiri 258	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
16	4, 15	jaoi 857	. . 3 ⊢ ((((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
17	3, 16	ax-mp 5	. 2 ⊢ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)
18		f1omo.1	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))
19	18	fveq1d 6888	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋))
20	19	eleq2d 2819	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
21	20	mobidv 2547	. 2 ⊢ (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
22	17, 21	mpbiri 258	1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847  ∀wal 1537   = wceq 1539   ∈ wcel 2107  ∃*wmo 2536  ∅c0 4313  {csn 4606   × cxp 5663  ‘cfv 6541  1_oc1o 8481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-1o 8488

This theorem is referenced by:  indthinc  49087  prsthinc  49089